18 problems

Equivalence conjecture. For every simple simply connected Lie group $G$ and every level $k\geq 0$, the categories $\mathrm{Rep}^k(LG)$ defined via textit{a.} or textit{b.}, and via…

Let $\mathcal A$ be a chiral conformal net with a finite gauge-group action by $\Gamma$, and let $\mathcal A^\Gamma$ denote its fixed-point orbifold net. Consider the irreducible D…

Let $W_k(\mathfrak{g},f)$ be a unitary minimal $W$-algebra, with corresponding conformal net denoted by $\mathcal{A}_{W_k(\mathfrak{g},f)}$. The algebra is strongly rational when i…

Compactness claim. The group $\pi_{0}\bigl(G\text{--}\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)$ is compact.

Let $\mathcal{A}$ be an irreducible graded-local conformal net, meaning a graded-local conformal net satisfying the irreducibility condition. Converse conformal-net conjecture. For…

Fix a modular tensor category $\mathcal C$ and a central charge $c$. Consider local conformal nets whose representation category is $\mathcal C$ and whose central charge is $c$. Fi…

A local conformal net is holomorphic when its only irreducible representation is the vacuum representation. Consider a holomorphic local conformal net with central charge $c=24$, a…

A completely rational local conformal net is a local conformal net with finite representation theory in the sense used in the source, and a simple unitary $C_2$-cofinite vertex ope…

Let $G$ be the compact, simply connected Lie group and $k$ the level appearing in the representation categories. Let $\mathrm{Rep}(\cA_{G,k})$ be the category of representations of…

For a completely rational conformal net $\mathcal{A}$, write $\operatorname{Rep}(\mathcal{A})^{\mathrm{rev}}$ for the unitary modular tensor category obtained by replacing its brai…

Let $\mathcal{F}$ be a unitary fusion category, and let $Z(\mathcal{F})$ denote its Drinfeld center. For a finite-depth subfactor $N\subset M$, let $D(N\subset M)$ denote its quant…

A unitary modular tensor category (UMTC) is a unitary braided fusion category with non-degenerate braiding. For a completely rational conformal net $\mathcal{A}$, let…

Let $G$ be a connected Lie group and let $k\in H^4(BG,\mathbb{Z})$ be a positive level. Consider the categories $\operatorname{Rep}^k(LG)$ defined via vertex operator algebras and…

Vertex algebra–conformal net correspondence conjecture. There is a bijective correspondence between completely rational local conformal nets and unitary $C_2$-cofinite vertex opera…

Lattice VOA conjecture.

Mirror extension conjecture. Under these hypotheses, rationality of the extension $U^e$ implies rationality of the mirror extension $(U^c)^e$.

Suppose that ${\cal A}\subset {\cal B}$ are conformal subnets with finite index. Let the vacuum representation of ${\cal B}$ carry the action of ${\cal A}$, and let $V$ be the spac…

Let $\mathit{Fer}(n)$ denote the conformal net of $n$ free fermions, and let $\mathit{CN}$ denote the 3-category of conformal nets, with its notion of $\mathit{CN}$-equivalence. Fo…